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Chapter 1: Problem 45

Multiply. \((\sin \theta-4)^{2}\)

Short Answer

Expert verified
The expansion of \((\sin \theta-4)^2\) is \(\sin^2 \theta - 8\sin \theta + 16\).

Step by step solution

01

Recognize the Formula

The exercise requires us to square a binomial expressed as \((\sin \theta - 4)^2\). We will use the expansion formula of a squared binomial: \((a - b)^2 = a^2 - 2ab + b^2\).
02

Identify Components

In the expression \((\sin \theta - 4)^2\), identify \(a = \sin \theta\) and \(b = 4\).
03

Apply the Formula

Substitute \(a = \sin \theta\) and \(b = 4\) into the formula: \((\sin \theta - 4)^2 = (\sin \theta)^2 - 2(\sin \theta)(4) + 4^2\).
04

Simplify the Expression

Calculate each part of the expression: \((\sin \theta)^2 = \sin^2 \theta\), \(-2(\sin \theta)(4) = -8\sin \theta\), and \(4^2 = 16\).
05

Combine Components

Combine the simplified terms: \(\sin^2 \theta - 8\sin \theta + 16\). This is the expanded form of \((\sin \theta - 4)^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Expansion
Binomial expansion is a powerful algebraic tool used to expand expressions of the form \((a+b)^n\) or \((a-b)^n\). For squaring a binomial, such as \((a-b)^2\), a specific formula simplifies this process: \[(a - b)^2 = a^2 - 2ab + b^2\]. This formula derives from distributing each part of the binomial across the other 鈥� essentially applying the distributive property.
  • The term \(a^2\) represents the square of the first term.
  • The term \(b^2\) represents the square of the second term.
  • Finally, \(-2ab\) accounts for twice the product of the two terms.

By memorizing these steps, one can quickly and accurately expand any binomial of the form \(a-b\), streamlining complicated algebraic tasks.
Squaring Trigonometric Expressions
Squaring trigonometric expressions involves raising trig functions to the power of two, most commonly using formulas like \((sin \theta)^2\). In the problem \((\sin \theta - 4)^2\), we apply what we know about trigonometric identities and squares:
  • First identify the components to be squared: \(\sin \theta\) is the trigonometric expression here.
  • Square each component separately as part of the binomial expansion: \((\sin \theta)^2 = \sin^2 \theta\).

When squared, trigonometric values like \(\sin^2 \theta\) often appear in identities, such as \(\sin^2 \theta + \cos^2 \theta = 1\), helping simplify expressions further when combined with other terms.
Simplification of Algebraic Expressions
Simplifying algebraic expressions involves a step-by-step process to make them less complex. This results in an expression that is easier to read and use. For the given expression \((\sin \theta)^2 - 8\sin \theta + 16\), basic simplification techniques include:
  • Combining like terms: Often we see like terms combined to reduce the expression, though in this specific case the terms are distinct.
  • Substitute applicable trigonometric identities if they simplify the expression further.
  • Check each step: Ensure each calculation follows algebraic rules, preventing potential errors.

The intent behind simplification is to convey mathematical information in its clearest form, aiding further calculations or analysis.

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Most popular questions from this chapter

Write each of the following in terms of \(\sin \theta\) only: \(\cos \theta\)

Add and subtract as indicated. Then simplify your answers if possible. Leave all answers in terms of \(\sin \theta\) and/or \(\cos \theta\). $$ \frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta} $$

Add and subtract as indicated. Then simplify your answers if possible. Leave all answers in terms of \(\sin \theta\) and/or \(\cos \theta\). $$ \cos \theta+\frac{1}{\sin \theta} $$

Find an angle \(\theta\) between \(0^{\circ}\) and \(360^{\circ}\) for which \(\cos \theta=-1\).

Find an angle \(\theta\) in the first quadrant for which \(\tan \theta=1\).
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